Birefringence of different artificial structures
1. Introduction
Birefringence is an important optical effect of materials that having different refractive indices for different polarizations of light. Birefringence and related optical effects play an important role in quantum and nonlinear processes and also have been widely used in modern optical devices, such as optical sensors, light modulators, liquid crystal displays, crystal filters, medical diagnostics, and wave plates (WPs). Among these, WP is one of the most essential elements in many optical modules and equipment, and will certainly have many applications in future photonic integrated circuits (PICs).
According to the difference of generation mechanism, birefringence could be divided into two types: natural birefringence and artificial birefringence, which are microscopic and macroscopic anisotropy induced, respectively [1]. Generally, artificial birefringence is larger than natural birefringence and has designable characteristics. Many artificial structures with high birefringence have been proposed and studied recently. Table 1 gives the comparison of birefringence, dispersion and loss of different structures containing multilayer film (MF) [2], nanowires (NW) [3], metamaterials (MM) [4], plasmonic nanoslits array (PNA) [1], multislotted dielectric waveguides (MSDW) [5], bulk photonic crystal (PhC) [68], two dimensional (2D) PhC waveguide (PhCW) [9, 10], and periodic dielectric waveguide (PDW) which also named as one dimensional (1D) PhCW [11, 12]. Among these artificial materials, the PhC related structures are more important not only for the excellent birefringence properties but also the great importance of PhC in PICs.
PhCs are structures with periodic arrangement of dielectrics or metals, which provide the ability of molding the flow of light in it [1315]. Due to the unique guiding properties of PhC structure, such as the photonic band gap guidance, it is foreseen as one of the key artificial materials for next generation PICs. This chapter gives a thorough review of the birefringence properties of PhC structures containing 1D PhCW (PDW), 2D PhCW and bulk PhC. The applications of the giant birefringence of PhC structures in both loworder WPs and highorder WPs are studied in details. This chapter is organized as follow. In Sec. 2, an overview of numerical algorithms used in this chapter is given. In Sec. 3, the birefringence properties are studied taking 2D PhCWs, 2D bulk PhCs, and 1D PhCWs as examples. High performance WPs are designed based on different PhC structures in Sec. 4. Discussions and conclusions are given in Sec. 5.
Δn  ~0.3  ~0.8  ~3.2  2.7  1.0  0.111  0.938  1.5 
Achromatic      No  No    Yes  Yes  Yes 
Loss  Low    High  High  Low  Low  High  Low 
* This table is from reference [12] 
2. Numerical algorithms
Many numerical algorithms which solve the partial differential equations can be used in computational photonics. For PhCs, two categories of problems are most important [15]: one is frequencydomain eigenvalue problem which refers to find the band structure
2.1. Plane Wave Expansion method (PWE)
PWE is used to solve the Maxwell equations by formulating an eigenvalue problem. The master equation for PhCs can be deduced from Maxwell equations as follow:
where
where
The PWE is suitable to calculate the band structure of PhC, but not conventional to get the transmission spectra of PhC. When the loss and transmitting properties of PhC are required, FDTD method is often used for the advantages of that broadband response can be accurately obtained in only one simulation run.
2.2. FiniteDifference TimeDomain method (FDTD)
The FDTD method is one of the gridbased differential timedomain numerical modeling methods. The timederivative parts of Maxwell equations in partial differential form are written:
where B is the electric displacement, D is the magnetic induction fields, J is the electriccharge current density, and J_{B} is an imaginary magneticcharge current density for calculation convenience. By centraldifference approximations, e.g. standard Yee grid [19], the electric and magnetic fields governed by Eq. (3) are discretized both in time and space. By properly selecting of initial excitation current J or J_{B}, the resulting finitedifference equations are solved in a leapfrog manner: the electric field vector components in a volume of space are solved at a given instant of time; then the magnetic field vector components in the same spatial volume are solved at the next instant of time; and the process is repeated over and over again until the desired transient or steadystate electromagnetic field behavior is fully evolved.
Although a lot of FDTD solver packages are available in literature, we use the free software MEEP [18] which is developed by MIT’s researchers in this chapter. The transmission spectrum of PhC structures can be easily obtained by the FDTD solver. Still take the square lattice PhCW studied in Fig. 1 as example, the transmission spectrum for the TM polarization of the PhCW with length of 27
2.3. Spatial Fourier Transform method (SFT)
Except for the low loss of the PhC structures, large birefringence is essential for realizing the ultracompact WPs. The frequencydependent effective mode indices for both TE and TM modes of PhC structure should be calculated to obtain the birefringence of the structures. When the PhC structure has single mode in the operating frequency range, such as PBG guided TM mode, in the PhCW shown in Fig. 1, the mode indices can be obtained from the dispersion curves calculated by PWE method. But, for the PhC structures with quasiperiodic and nonperiodic structures or some special PhCW without PBG guided modes but total internal reflection (TIR) modes, the dispersion curves of the guided modes are difficultly obtained by the conventional PWE method. In this chapter, the SFT method [20] is used when the birefringence of this category of structure needs to be calculated.
The SFT method is based on the spatial Fourier transform spectrum of the electromagnetic field distributions of the waveguide mode along the propagating direction. Assuming u_{ω}(
To verify the SFT method, the field distribution of TM mode in the PhCW studied in Fig. 1 and Fig.2 are calculated by FDTD method and the snapshot of E_{z} fields at the frequency of 0.40
3. Birefringence of PhC structures
Birefringence is related to the effective index difference of two orthogonal polarization modes and can be expressed as Δ
3.1. Birefringence of 2D PhCW
For birefringence related applications, both TE and TM polarized light must propagate with low loss in the 2D PhCW which is formed by introducing of line defect in the perfect bulk PhC at a given direction. Different guided mechanisms could be used to confine light in the 2D PhCWs [21]. The widely studied mechanism in the literature is the PBG guiding as that shown in Fig. 1. It is difficult to realize 2D PhCW supporting both TE and TM PBG guided modes, only if carefully choosing the materials and geometry structures. Studied results show that the 2D PhCW can also guide the light as that in the conventional dielectric slab waveguide by total internal refection (TIR) guiding mechanism. Compound of PBG and TIR effects could also make low loss guiding for both TE and TM mode in the 2D PhCW [2123].
3.1.1. Birefringence of 2D PhCW with PBG guided modes
The PhCW with hybrid triangular and honeycomb lattices can support both TE and TM modes [24, 25]. The structure is shown in Fig. 4 and has been optimized as follow: The permittivity of the background high index material is 11.56, and the radii of the large and small air holes are
The band structure of PhCW shown in Fig. 4 is calculated by PWE method with 1×4
3.1.2. Birefringence of 2D PhCW with hybrid PBG and TIR guided modes
The PhCWs with both square and triangular lattice air holes in high index materials can support low loss transmitting of TE and TM polarizations with the help of TIR guided mechanism [2123]. Taking a square lattice PhCW as example, the waveguide is formed by introducing of a line defect in ΓX direction in the perfect PhC which has square lattice air holes in high index material with permittivity of 12.96. Calculated dispersion curves of the PhCW are shown in Fig. 6 for both TE and TM polarizations. The PhCW has single TE guided mode in the normalized frequency range of 0.2480.272
3.2. Birefringence of 2D bulk PhCs
Actually, the typical bulk PhC itself is strongly anisotropic artificial material which provides large birefringence. Different from the PhCW, the PhC as birefringent media must work outside the PBG for that the PhC are highly reflection material for the EM wave located in the PBG frequency range. The birefringence of bulk PhC is measured for the hexagonal lattice structure in microwave band and the experimental measurement birefringence is below 0.20 [8]. By increasing the index difference of the materials, the birefringence of the bulk PhC could be much stronger. Taking a bulk 2D structure as example, the PhC is composed by parallel cylinder dielectric rods in air, in which the dielectric rods have radius of
For the birefringence related applications, the bulk PhC is a good candidate for the large birefringence in it, however, there are still some disadvantages [27]: 1) Lacking of effective light confining in the propagating plane makes beam divergence, as shown in Fig. 8 (a), which will spread the EM fields into the adjacent devices and cause crosstalk if there are many components packaged compactly, such as in PIC, to fulfill complicated functions. 2) The attenuation of light in the bulk PhC is high for the high scattering loss in it. These two problems should be solved in the practical applications such as WPs.
3.3. Birefringence of 1D PhCW
As shown in Fig. 9 (a), the 1D PhCW here refers in particular to the PDW [2830], also known as nanopillar periodic waveguide [3135] or coupled periodic waveguide [36], which has periodicity only in the light propagating direction. The 1D PhCW has attracted a lot of research interests for the simpler structure comparing with the 2D PhCW, and can be used in slow light [37], laser [35], lowloss waveguide [2830, 38, 39], microresonator cavities [40], splitters for polarization and frequency [4144], etc. The dispersion curves of the 1D PhCW could be examined by PWE method as what shown in Fig. 9 (b). There are guided modes under the light line and only single TE and TM modes are supported by the structure when the frequency is under 0.2065
The birefringence properties of the 1D PhCW with cylinder dielectric rods in air are shown in Fig. 10, and all birefringence values are calculated in the single mode frequency band. The 1D PhCW has giant birefringence, which is larger than 1.5, when the permittivity of dielectric rods is 12.96. The higher the permittivity of dielectric rods is, the larger the birefringence is. The birefringence varies rapidly in the frequency band nearby the upper edge (slow light region) of the first TM guided mode when the 1D PhCW having relatively small dielectric rods (e.g.
Other types of 1D PhCW have large birefringence too. Taking the 1D PhCW with square air holes in dielectric waveguide as example, the birefringence is around 1 when the length of side of the square is
4. Ultracompact WPs based on PhC structures
One of the most important devices for birefringence related applications is WP which is worked as phase retarder. The phase difference (Δ
WPs with broadband achromatic phase difference are widely used in practice for the most of polarizationphase controlling devices require frequency independent phase retarding. Beyond that, compact in size is essential for that the original intention is realizing phase retarding in PICs for PhC structures based WPs. From the expression of the phase difference, the value of Δ
For the large and designable of birefringence in PhC structures, high performance WPs can be realized, such as loworder broadband achromatic and highorder compact QWPs and HWPs. This section will focus on the loworder WPs based on 2D PhCW, 2D bulk PhC and 1D PhCW, respectively, and the compact highorder WPs based on the so called formed birefringence are discussed, too.
4.1. Loworder WPs based on 2D PhCW
For the simplicity of the 2D PhCW with hybrid PBG and TIR modes, the square lattice air holes type of PhCW studied in Fig. 6 and Fig. 7 is used to design the loworder WPs. The structure used in the FDTD simulation is shown in Fig. 11 (a). A waveguide broadband Gaussian pulse source with width of
4.2. Loworder WPs based on bulk 2D PhCs
The bulk 2D PhC has larger birefringence than the 2D PhCW, therefore, the WPs realized by bulk PhC has smaller size. However, the beam divergence is severe in the perfect PhC structure as discussed before and the divergent beam in nonwaveguiding structure will interfere with other components for large number of devices integrated in ultrasmall space in practical PICs [27]. Interference and scattering loss caused by beam divergence can be solved by the so called selfcollimating (SC) effect under which light beam can propagate with no diffraction in perfect PhCs [5052].
Polarization independent SC propagation is need in PhC for the WP applications. As shown in the inset of Fig.12 (a), the designed PhC is square lattice air holes in dielectric materials, and the permittivity of host material and radius of air holes are
By launching broadband Gaussian pulse source in the coupling waveguide (refer Fig. 12 (d)), the transmission behaviors of the SC beam in the PhC are quantified and the spectra are shown in Fig. 13 (a) and (b) for TE and TM polarization, respectively. In the polarization independent SC band, the transmission efficiencies are above 75% for both TE and TM polarizations when the length is shorter than 70
To verify the phase characteristics of the WPs designed above, a CW source is launched in the dielectric waveguide, and a set of point monitors are inserted along the
4.3. Loworder WPs based on 1D PhCW
Comparing with the 2D PhCW, 1D PhCW has larger birefringence, so that it is very suitable for the ultracompact loworder WP applications. Both 1D PhCW with dielectric rods in air and air holes in dielectric waveguides have been used to design high performance WPs.
4.3.1. Loworder WPs based on 1D PhCW with dielectric rods in air
The birefringence properties of 1D PhCW with dielectric rods in air have been studied thoroughly in section 3.3. Although there is broadband achromatic birefringence, it is not enough to achieve wide band achromatic phase difference for Δ
For the ultrahigh birefringence, 2
Except for the waveguide dispersion, the material dispersion also can affect the effective indices of the waveguide modes, and then affect the birefringence. To study the phase difference of 1D PhCW with dispersive material, Silicon (Si) is chosen as the material of dielectric rods for the permittivity of Si can be obtained by Sellmeier formula:
where the wavelength λ is in μm. The radius of the Si rods is 0.55a. The phase difference (Δ
4.3.2. Loworder WPs based on 1D PhCW with air holes in dielectric waveguide
Another type of 1D PhCW is the periodic air holes in dielectric waveguide as shown in Fig. 9 (c) and (d). This type of structure is easier integrated with other components for the most of devices in PIC are constructed and interconnected by waveguide. Here, the 1D PhCW with square air holes is used to design high performance loworder WPs. The width of the waveguide is as same as lattice constant
The phase differences between TE and TM polarizations are directly studied by FDTD simulation method. A CW source with frequency of
The shortage of the structure studied above, referring to nontaper structure, is that the transmission loss is relatively high as shown in Fig. 18 (a) and (b).To reduce the loss, a taper structure is used to design high performance WP and the structure is shown at the top of Fig. 18.The transmission efficiencies are effectively improved by the taper 1D PhCW with
4.4. Highorder WPs based on formed birefringence effect
Except for the loworder achromatic WPs, highorder WPs are also useful in some special applications. By increasing the length of the PhC structures, highorder WPs can be realized, but it is not inadvisable when the birefringence is not large enough and the loss increases rapidly with the increasing of length. Another method to realize highorder WPs proposed before is socalled formed birefringence which takes full advantage of birefringence and optical path difference between TE and TM polarizations [27, 10, 12].The schematic diagram is shown in Fig. 19 (a). The key point of the formed birefringence is bringing path difference of two orthogonal polarizations into use to enhance the phase difference between them. As that shown in Fig. 19 (a), the lengths of path for polarization 1 (TE or TM) and polarization 2 (TM or TE) are denoted as
where
where Δ
The highorder WP by formed birefringence effect is designed based on dielectric rod type of 1D PhCW with
Except for the 1D PhCW, the bulk PhC with polarization independent SC effect and the 2D PhCW are also can be used to design highorder WPs based on formed birefringence effect. The polarization beam splitter and combiner can be realized by direct coupler in 2D PhCW structure, and by reflecting and transmitting mirror in bulk PhC structure. Actually, the 80^{th} order WP has been designed by the 2D PhCW with triangular lattice air holes in high index dielectric material [10].
5. Conclusion
In this chapter, the birefringence properties of three types of PhC structures, containing 2D PhCW, 2D bulk PhC, and 1D PhCW, have been studied thoroughly. High performance WPs based on the birefringence of these three types of PhC structures have been proposed. The comprehensive remarks about the three PhC structures are shown in Table 2.
For the 2D PhCW, the birefringence is not very high, so that the size of the WPs based on it is in wavelength magnitude. Although the achromatic bandwidth is not very large, 2D the PhCW provide perfect guiding for the light in it.
For the 2D bulk PhC, the birefringence in it is much larger than that in 2D PhCW. The disadvantage of the bulk PhC is the beam divergence which will cause scattering loss and signal crosstalk. This problem has been improved by the SC effect in this chapter. The WPs based on the 2D PhC with polarization independent SC effect have compact size and broad achromatic bandwidth (about 45nm).
For the 1D PhCW, giant birefringence, which is even larger than 1.5, can be realized in special structures. The 1D PhCW is very suitable for WP applications for its giant birefringence, low loss and compact in size. The achromatic bandwidth of the WPs based on 1D PhCW can be larger than 100nm with excellent phase accuracy of ±1°. Meanwhile, the size of the WP is in subwavelength magnitude.
Besides, the highorder WPs based on so called formed birefringence is proposed too. Additional phase differences are introduced by the path difference of different polarizations. The 500^{th} order WP is designed based on the formed birefringence by using the 1D PhCW.



2D PhCW  ~0.1  ~6nm (±0.005 
80  3.7λ  Low 
Bulk PhC  ~0.8  ~45nm (±0.01 
    Relatively high 
1D PhCW  ~1.5  "/>100nm (±1°)  500  0.67λ  Low 
The WP is one of the basic elements in optical devices. The proposed PhC structure based WPs have a lot potential applications in future PICs for sensing, optical communications and measurements.
Acknowledgments
The authors would like to gratefully acknowledge W. –P. Huang at McMaster University, J. –W. Mu at MIT and J. Liu at Xi’an University of Posts and Telecommunications for their fruitful suggestion. The study was supported by CAS/SAFEA International Partnership Program for Creative Research Teams, West Light Foundation of The Chinese Academy of Sciences, and National Natural Science Foundation of China (Grant No. 61275062).
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